Section 2 : The Binomial Distribution(4)
 

[cumulative probability tables - the case of p>0.5]

[ P(X<x) ][ P(X=x) ][ P(X<x) ][ P(X>x) ][ P(X>x) ]

 

 

 

 

Cumulative probability tables - case of p>0.5

These give the tabulated value of P(X< x) . This means that the probability displayed is less than or equal to an observed value of x.
The random variable X is distributed Binomially, where there are n trials and probability of success p .

Working out values of random variable probabilty P(X) for the case of p>0.5 is complicated by the fact that values of p only go up to 0.5 .

The way around this problem is to consider another random variable Y , representing failure.

So we have:                      pX + pY = 1

In the same way as X, Y is distributed binomially:

Y ~ B(n,1 - pY)

Say that the random variable X has values X = 0, 1, 2, 3, 4, 5

A table of values for X (success) and Y (failure) looks like this:

X
0
1
2
3
4
5
Y
5
4
3
2
1
0

 

The method is to use the table to produce an expression in Y that will use values of p<0.5 .

back to top

 

The case      P(X<x)

Say we wish to find the value of P(X<4) for :

X = 0, 1, 2, 3, 4, 5      pX=0.85*       n=6

*tables only go up to 0.5

X (success) and Y (failure) are related so:

X
0
1
2
3
4
5
Y
5
4
3
2
1
0

The sum of successes and failures for each outcome must always be the same (ie 5).

The probability for Y becomes py=0.15*     (pX + pY= 1)

* py <0.5 and therefore on the table

From the table,

P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)

= P(Y=5) + P(Y=4) + P(Y=3) + P(Y=2) + P(Y=1)

this can be written:

P(X<4) = P(Y>1)

since,

P(Y=0) + P(Y=1) + P(Y=2) + P(Y=3) + P(Y=4) + P(Y=5) = 1

P(Y=1) + P(Y=2) + P(Y=3) + P(Y=4) + P(Y=5) = 1 - P(Y=0)

in other words,

P(Y>1) = 1 - P(Y<0)

hence the original inequality can be rewritten :

P(X<4) = 1 - P(Y<0)

Using the tables to find the value of P(Y<0) for n=6 pY=0.15 , Y=0 :

P(Y<0) = 0.3771

binomial distribution - cumulative probability table #2

hence,

P(X<4) = 1 - 0.3771 = 0.6229

 

back to top

 

The case      P(X=x)

Consider the binomial distribution for success,

X ~ B(n, pX)

X = 0, 1, 2, 3, 4, 5      pX=0.85       n=6

also the binomial distribution for failure,

Y ~ B(n, pY)

Y= 0, 1, 2, 3, 4, 5      pY=0.15       n=6

 

Say we want to find P(X=4).

X
0
1
2
3
4
5
Y
5
4
3
2
1
0

From the table it follows that,

P(X=4) = P(Y=1)

P(Y=1) = P(Y<1) - P(Y<0)

P(X=4) = P(Y<1) - P(Y<0)

P(X=4) = 0.7765 - 0.3771 = 0.3994

 

back to top

 

The case     P(X<x)

Consider the binomial distribution for success,

X ~ B(n, pX)

X = 0, 1, 2, 3, 4, 5      pX=0.85       n=6

also the binomial distribution for failure,

Y ~ B(n, pY)

Y= 0, 1, 2, 3, 4, 5      pY=0.15       n=6

 

Say we want to find P(X<4).

X
0
1
2
3
4
5
Y
5
4
3
2
1
0

From the table it follows that,

P(X<4) = P(Y>1)

and

P(Y>1) = P(Y=2) + P(Y=3) + P(Y=4) + P(Y=5)

it follows that,

P(Y>1) = P(Y>2)

P(Y>2) = P(Y=2) + P(Y=3) + P(Y=4) + P(Y=5)

P(Y=0) + P(Y=1) + P(Y=2) + P(Y=3) + P(Y=4) + P(Y=5) = 1

P(Y=2) + P(Y=3) + P(Y=4) + P(Y=5) = 1 - [ P(Y=0) + P(Y=1)]

P(Y>2) = 1 - [ P(Y=0) + P(Y=1)]

P(Y>2) = 1 - P(Y<1)

P(X<4) = 1 - P(Y<1)

P(X<4) = 1 - 0.7765 = 0.2235

 

back to top

 

The case     P(X>x)

Consider the binomial distribution for success,

X ~ B(n, pX)

X = 0, 1, 2, 3, 4, 5      pX=0.85       n=6

also the binomial distribution for failure,

Y ~ B(n, pY)

Y = 0, 1, 2, 3, 4, 5      pY=0.15       n=6

Say we want to find P(X>4).

X
0
1
2
3
4
5
Y
5
4
3
2
1
0

P(X>4) = P(Y<1)

P(Y<1) = P(Y=0)

P(Y=0) = P(Y<0)

P(X>4) = P(Y<0)

reading P(Y<0) directly from the tables,

P(X>4) = 0.3771

 

back to top

 

The case     P(X>x)

Consider the binomial distribution for success,

X ~ B(n, pX)

X = 0, 1, 2, 3, 4, 5      pX=0.85       n=6

also the binomial distribution for failure,

Y ~ B(n, pY)

Y = 0, 1, 2, 3, 4, 5      pY=0.15       n=6

Say we want to find P(X>4).

X
0
1
2
3
4
5
Y
5
4
3
2
1
0

P(X>4) = P(Y<1)

reading P(Y<1) directly from the tables,

P(X>4) = 0.7765

 

back to top

 

VIDEO

the mean
the median
stand. deviation 1
stand. deviation 2
stand. deviation 3
z-scores
confidence interval
goodness of fit
distrib. sample mean
t interval
chi-squared test
 
more...Video Library
 

INTERACTIVE

normal distribution
mean,median comprd 1
mean,median comprd 2
type I & II errors
linear regression
histogram,box whisker
 
 

EXAM PAPERS(.pdf)

Edxl S1 Statistics spec.
Edxl S1 Statistics ans.
Edxl S2 Statistics spec.
Edxl S2 Statistics ans.
Edxl S3 Statistics spec.
Edxl S3 Statistics ans.
 

TOPIC NOTES(.pdf)

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Google
your stop for the best in math, science & programming tutorials on the Net revision help to get a better result incremental success advanced physics for secondary/high school, including much in-depth content common to first year university courses your one stop for the best in math, science and programming tuition revision help for a better result incremental success advanced physics for high school/secondary and 1st year university fast-track learning for everyone

[ PURE MATHS ][ MECHANICS ][ STATISTICS ]